Chapter One: Dimensions, Units, and Their Conversion

Transcription

1 Lecture # 1, 2/8/2012 Chapter One: Dimensions, Units, and Their Conversion Objectives: 1. Understand and explain the difference between dimensions and units. 2. Add, subtract, multiply and divide units associated with numbers. 3. Specify the basic and derived units in SI and AE systems. 4. Convert one set of units in an equation into another set. 5. Apply the concepts of dimensional consistency. 6. Employ an appropriate number of significant figures. 1

2 Units and Dimensions Dimensions: The basic concepts of measurement such as length, time, mass. Units: Method of expressing the dimensions such as feet, centimeters. Quantity = numerical value & units Example : 2 meters 1/3 second 2.29 kilograms A dimension is a property that can be: measured such as length, mass or temperature, or calculated by multiplying or dividing other dimensions such as length/time (velocity). 2

3 Units and Dimensions The two commonly used systems of units: SI: Le Systeme Internationale d Unities Length = meter (m) Mass = kilograms (kg) Temperature = Kelvin (K) or Celsius (C) Time = second (s) AE: American Engineering system Length = foot (ft) Mass = pound-mass (lb m ) Temperature = Fahrenheit (F) or Rankine (R) Dimensions and their units are classified as: Fundamental: Units that can be measured independently and defined by convention such as grams for mass, seconds for time. Derived: Units developed in terms of the fundamental units. They can be obtained by multiplying or dividing fundamental units such as cubing length results in volume (liter). 3

4 Table1.1: SI Units 4

5 Table 1.2. AE Units 5

6 Unit abbreviations The distinction between uppercase and lowercase should be followed, example: kg, K. They have the same form for both the singular and plural, example: 1 m, 10 m. They are not followed by a period (except for inches, in.) A compound unit, which is formed by multiplication of two or more other units, consists of the symbols for the separate units joined by a dot. (example: N.m), except in the case of familiar units such as watt-hour (Wh) and if the symbols are separated by exponents such as N.m -2 kg -2. 6

7 Table 1.3: SI Prefixes For SI system, units and their multiples and submultiples are related by standard factors designated by the prefix (except for time). 7

8 Operations with Units Addition and Subtraction of quantities You can add or subtract numerical quantities only if the units are the same 5 kilograms + 3 joules 10 pounds + 5 grams Multiplication and Division of quantities You can multiply or divide unlike units, but cannot cancel units unless they are identical 50 (kg)(m)/(s) 3 m 2 /60 cm 3 m 2 /0.6 m 5 m 8

9 Conversion of units The procedure for converting one set of units to another is simply to multiply any number and its associated units by ratios termed conversion factors to arrive at the desired answer and its associated units. Useful conversion factors can be found inside the front cover of the text book. 9

10 Examples Convert the following number to desired units: a. 10 cm to in. b. 1.8 nm to dm c. 42 ft 2 /hr to cm 2 /s d cal/(gmol)(k) to Btu/(lb mol)(⁰r) 10

11 Test Yourself: 1. Convert 1 cm/s 2 to km/yr 2 2. Convert 23 lb m.ft/min 2 to kg.cm/s 2 1lb m =0.453 kg 1m=3.281 ft 11

12 Dimensional Consistency (Homogeneity) A basic principle stated that equations must be dimensionally consistent, which means, each term in an equation must have the same net dimensions and units as every other term to which it is added, subtracted, or equated. Consider this equation: u = u₀ + g.t In SI system: u(m/s) = u₀(m/s) + g(m/s 2 )t(s) This equation is dimensionally homogeneous, since each of the term u, u₀, and gt has the same dimensions (length/time) 12

13 Example: Dimensional Consistency Consider the equation D(ft) = 3t(s) + 4 What are the dimensions and units of 3 and 4? Derive an equation for distance in meters in terms of time in minute? 13

14 Test yourself: equation consistency A crystal of diameter D (mm) is placed in a solution of dissolved salt, and new crystals are formed at a constant rate r (crystals/min). Experiments show that the rate of crystal formation varies with the crystal diameter as: r ( crystals/min)= 200 D-10D 2 (D in mm) a) What are the units of the constants 200 and 10? b) Calculate the crystal formation rate in crystal/s for a crystal diameter of in. c) Derive a formula for r (crystals/s) in terms of D( inches). 14

15 Dimensionless Groups Dimensionless groups are used extensively in Chemical Engineering to determine relationships between parameters, either by theory or based on experiment. They have no units. Reynolds Number (Re): Reynolds number is influenced by fluid properties (viscosity and density), flow conditions (velocity) and geometry (relevant length scale). Reynolds number is the key parameter for determining whether flow is Laminar or Turbulent. Re = ρvl/μ or ρ- Density (kg/m 3 in SI Units) V - Velocity (m/s) L - Length Scale (such as diameter of pipe) (m) μ - Dynamic Viscosity (Pa.s) or (kg/m.s) 15

16 Test yourself: Dimensionless Groups Calculate the Re for the following set of conditions for a fluid flowing in a pipe: Re = ρvd/μ ρ- Density = 62.3 lb/ft 3 1 lb=454 gr V - Fluid Velocity = 2 m/s 1 m= ft D - Pipe Diameter = 2 in. 1 in.=2.54 cm μ - Viscosity = 2.5 g/cm.s 16

17 Force According to Newton s 2nd Law of Motion, force is proportional to the product of mass and acceleration (length/time 2 ), F= m.a In the SI system, the unit of force is defined to be the Newton (N) when 1 kg is accelerated at 1 m/s 2. m = 1 kg a = 1 m/s 2 F = 1 N 1 newton (N) 1 kg.m/s 2 F= Cma = 1N 1kg 1 m = 1N C= 1 (kg)(m) s Conversion factor 2 s 2 N kg.m s 2 In the American engineering system, the derived force unit is called a pound-force (lb f ) and is defined as the product of a unit mass (1 lb m ) and the average acceleration of gravity at sea level at 45 latitude, which is ft/s 2 (depending on the location of the mass) m = 1 lb m a =g= ft/s 2 F = 1 lb f 1 lb f lb m.ft/s 2 F= Cma = 1 lb f 1 lb m g ft = 1 lb f lb m.ft /s 2 s 2 g c = 1/C= 1 kg.m/s 2 = ft.lb m /s 2 1 N 1 lb f 17

18 Weight The weight of an object is the force exerted on the object by gravitational attraction. W = m.g The value of g at sea level and 45 latitude for each system of units is: g = ft/s 2 (AE) = m/s 2 (SI) 18

19 Example: weight Water has a density of 62.4 lb m /ft 3. How much does 2 ft 3 of water weigh (1) at sea level and 45 latitude and (2) in a location where the attitude is 5374 ft and the gravitational acceleration is ft/s 2? 19

20 Test yourself: Mass and Weight A waste treatment pond is 50 m long and 15 m wide, and has an average depth of 2 m. The density of the waste is 85.3 lb m /ft 3. Calculate the weight of the pond contents in lb f. 20

21 The Importance of Proper Units SEPTEMBER 30, 1999 Likely Cause Of Orbiter Loss Found The peer review preliminary findings indicate that one team used English units (e.g., inches, feet and pounds) while the other used metric units for a key spacecraft operation. Mars Climate Orbiter 21

22 The Importance of Proper Units Cont d Gimli Glider is the nickname of an Air Canada aircraft which was involved in an infamous aviation incident. On 23 July 1983, a Boeing jet, Air Canada Flight 143, ran completely out of fuel at 41,000 feet (12,500 m), about halfway through its flight from Montreal to Edmonton. The subsequent investigation revealed that fuel loading was miscalculated through misunderstanding of the recently adopted metric system which replaced the Imperial system. At 41,000 feet, over Red Lake, Ontario, the aircraft's cockpit warning system sounded, indicating a fuel pressure problem on the aircraft's left side. Assuming that a fuel pump had failed, the pilots turned it off, as gravity would still feed fuel to the aircraft's two engines. The aircraft's computer indicated that there was still sufficient fuel for the flight, but, as subsequently realized, the calculation was based on incorrect settings. A few moments later, a second fuel pressure alarm sounded, prompting the pilots to divert to Winnipeg. Within seconds, the left engine failed and they began preparing for a single-engine landing. As they communicated their intentions to controllers in Winnipeg and tried to restart the left engine, the cockpit warning system sounded again, this time with a long "bong" that no one present could recall having heard before. This was the "all engines out" sound, an event that had never been simulated during training. Seconds later, most of the instrument panels in the cockpit went blank as the right-side engine also stopped and the 767 lost all power. At this point, the pilot proposed his former airforce base at Gimli as a landing site. Unknown to him, however, the base had become a dragstrip and had decommissioned one of its runways. As a result of the runway's conversion to use as a dragstrip, the runway had been converted into two lanes with a guard rail running down the middle of it. Furthermore, a "Family Day" was underway at the dragstrip that particular day and the area around the decommissioned runway was covered with cars and campers. The decommissioned runway itself was being used to stage a race. As soon as the wheels touched the runway, the pilot "stood on the brakes, blowing out two of the aircraft's tires. The unlocked nose wheel collapsed and was forced back into its housing, causing the aircraft's nose to scrape along the ground. The plane slammed into a guard rail which made the plane lose a bit more speed to stop it from flying off the runway. None of the 61 passengers was seriously hurt during the landing. 22

23 Example: Units and Dimensions An empirical equation for calculating the inside heat transfer coefficient, h i, for the turbulent flow of liquids in a pipe is given by: G h i = D where h i = heat transfer coefficient, Btu/(hr)(ft) 2 ( F) G = mass velocity of the liquid, lb m /(hr)(ft) 2 K = thermal conductivity of the liquid, Btu/(hr)(ft)( F) Cp = heat capacity of the liquid, Btu/(lb m )( F) μ = Viscosity of the liquid, lb m /(ft)(hr) D = inside diameter of the pipe, (ft) a. Verify if the equation is dimensionally consistent. b. What will be the value of the constant, given as 0.023, if all the variables in the equation are inserted in SI units and h i is in SI units K µ Cp

24 Test yourself: Conversion of Units The Dittus-Boelter equation can be used to calculate the heat transfer coefficient in which a flowing fluid in a pipe is being heated (or cooled). The correlation involves physical properties of the fluid, and system dimensions, and is expressed as: hd/k = Re 0.8 Pr 0.4 where Re = Reynolds Number = ρvd/μ Pr = Prandtl Number = Cpμ/k Find the heat transfer coefficient, h, in Btu/h.ft 2. F and cal/s.m 2. C for the following conditions: D = pipe diameter = 2 inches k = thermal conductivity of the fluid = Btu/h.ft. F ρ = density of fluid = 55.2 lb/ft 3 v = velocity of fluid = 0.4 m/s μ = viscosity of fluid = 1.60 lb/ft.h Cp = heat capacity = Btu/lb. F 1 in= ft 1 m= ft 1 Btu=252 cal 1 C= 1.8 F 24

25 Significant Figures The significant figures of a number are the digits from the first nonzero digit on the left to either: (a) the last nonzero digit of the number if there is no decimal point, or (b) the last digit (zero or non-zero) on the right if there is a decimal point. The number of significant figures is readily seen if scientific notation is used. Scientific notation is generally a more convenient way of representing large or small numbers. Scientific notation: 1.23 x 10 8 Conventional number: 123,000,000 3 significant figures 25

26 Example: Significant Figures How many significant figures in the following? Re-express the numbers in scientific notation = 3 significant figures = 3.21 X = 7 significant figures = X = 4 significant figures = X

27 Significant Figures Multiplication and Division: The number of significant figures in your answer should equal the lowest number of significant figures of any of the numbers being multiplied or divided (3.57) (4.286) = ( ) ( )/(2.67) = = 320 Addition and Subtraction: Count the number of significant figures in the decimal portion of each number in the problem. The answer contains no more decimal places than the least accurate number = =